We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of correctness and proof normalization. We have implemented our work in the Isabelle Logical Framework.

1. What are called ‘intuitions ’ in philosophy are just applications of our ordinary capacities for judgement. We think of them as intuitions when a special kind of scepticism about those capacities is salient. 2. Like scepticism about perception, scepticism about judgement pressures us into conceiving our evidence as facts about our internal psychological states: here, facts about our conscious inclinations to make judgements about some topic rather than facts about the topic itself. But the pressure should be resisted, for it rests on bad epistemology: specifically, on an impossible ideal of unproblematically identifiable evidence. 3. Our resistance to scepticism about judgement is not simply epistemic conservativism, for we resist it on behalf of others as well as ourselves. A reason is needed for thinking that beliefs tend to be true. 4. Evolutionary explanations of the tendency assume what they should explain. Explanations that appeal to constraints on the determination of reference are more promising. Davidson’s truth-maximizing principle of charity is examined but rejected. 5. An alternative principle is defended on which the nature of reference is to maximize knowledge rather than truth. It is related to an externalist conception of mind on which knowing is the central mental state. 6. The knowledge-maximizing

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The course will study selected extensions of and alternatives to classical propositional logic. The extensions include modal, conditional, and tense logics; the alternatives include intuitionistic, many valued, and relevance logics. In each case, we will consider the philosophical issues raised by these logics, though most of our time will be devoted to becoming familiar with the logics themselves, from both a semantic and a proof-theoretic standpoint. Throughout the course, we will trade depth for breadth; the goal is not to study any of these logics in mathematical detail, but just to get a sense of the range of possibilities. Time and place

The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning". What I have to tell here is not a new story, and it does not contain any really new ideas. The main difference with my earlier discussions of the same topics ([TD88, chapter16],[Tro91]) is in the emphasis. This paper starts with some examples of the transition from informal concepts to mathematically precise notions, followed by a more detailed discussion of one of these examples, the intuitionistic notion of a choice sequence, arguing for the lasting interest of this notion for the philosophy of mathematics. In a final section, I describe my own position relative to some of the philosophical discussions concerning intuitionistic logic in the wr...

This paper is an attempt to understand the differences between them. I am grateful to Mitsuru Yasuhara for stimulating discussions of this material and for pinpointing errors and obscurities in earlier drafts. I also wish to thank Simon Kochen and Per Martin-Lof for helpful comments.

this dissertation I address both questions of application and questions of constitution. The primary aim of this chapter is to begin by addressing the question of what it means for judgments to be objective or subjective. One common use of the notions of objectivity and subjectivity is to demarcate kinds of judgment (or thought or belief). On such a usage, prototypically objective judgments concern matters of empirical and mathematical fact such as the moon has no atmosphere and two and two are four. In contrast, prototypically subjective judgments concern matters of value and preference such as Mozart is better than Bach and vanilla ice cream with ketchup is disgusting. I offer these examples not to take sides on whether such judgments actually are objective or subjective, but only to call attention to a typical way of using "objective" and "subjective". The question arises as to what it means in this context to call these respective judgments "objective" and "subjective". Some have proposed that the difference hinges on truth. Objective judgments are absolutely true, whereas the truth of subjective judgments is relative to the person making the judgment: my judgments are true for me, your judgments are true for you. You and I can each utter "vanilla tastes great" but in your mouth this may constitute a truth and in my mouth it may constitute a falsehood. Subjective judgments are subject relative. Some philosophers have noted an analogy between this kind of subject relativity and a kind that obtains for indexical expressions. You and I can both utter "I am here" and thereby express different propositions. Some philosophers have construed indexicality as an instance of subjectivity and some others have even gone so far as to argue that subjectivity just is indexicality....

Abstract: Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution. Part I The Problem §1. The epistemic conception of truth. The epistemic conception of truth can be formulated in many ways. But the basic idea is that truth is explained in terms of epistemic notions, like experience, argument, proof, knowledge, etc. One way of formulating this idea is by saying that truth and knowability coincide, i.e. for every statement S

It is not unreasonable to think that the dispute between classical and intuitionistic mathematics might be unresolvable or 'faultless', in the sense of there being no objective way to settle it. If so, we would have a pretty case of relativism. In this note I argue, however, that there is in fact not even disagreement in any interesting sense, let alone a faultless one, in spite of appearances and claims to the contrary. A position I call classical pluralism is sketched, intended to provide a coherent methodological stance towards the issue. Some reasons to recommend this stance are given, as well as some speculations as to why not everyone might want to follow the recommendation. 1.

Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl’s third claim is wrong, by his own standards. To explain this thesis, let me first introduce the term ‘revisionism’. It is understood here, following Crispin Wright, as the term that applies to ‘any philosophical standpoint which reserves the potential right to sanction or modify pure mathematical practice ’ [Wright 1980, p.117]. I want to make a distinction between weak and strong revisionism. The point of reference is the actual practice of mathematics. Weak revisionism then potentially sanctions a subset of this practice, while strong revisionism potentially not only limits but extends it, in different directions. In strong revisionism, certain combinations of limitation and extension may lead to a mathematics that is no longer compatible with the unrevised one. ‘May lead’, not ‘necessarily leads’: it is all a matter of reserving rights; whether there is occasion to exercise them is a further question. To illustrate these categories, let me give examples of non-revisionism, weak revisionism, and strong revisionism. Non-revisionism can be found in Wittgenstein’s Philosophische Untersuchungen, where philosophy can neither change nor ground mathematics: Die Philosophie darf den tatsächlichen Gebrauch der Sprache in keiner Weise antasten, sie kann ihn am Ende also nur beschreiben. Denn sie kann ihn auch nicht begründen. Sie läßt alles wie es ist. Sie läßt auch die Mathematik wie sie ist, und keine mathematische